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Gibbs Sampling Methods for StickBreaking Priors
"... ... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stickbreaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling meth ..."
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Cited by 222 (17 self)
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... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stickbreaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies to stickbreaking priors with a known P'olya urn characterization; that is priors with an explicit and simple prediction rule. Our second method, the blocked Gibbs sampler, is based on a entirely different approach that works by directly sampling values from the posterior of the random measure. The blocked Gibbs sampler can be viewed as a more general approach as it works without requiring an explicit prediction rule. We find that the blocked Gibbs avoids some of the limitations seen with the Polya urn approach and should be simpler for nonexperts to use.
Sequential Importance Sampling for Nonparametric Bayes Models: The Next Generation
 Journal of Statistics
, 1998
"... this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining t ..."
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Cited by 70 (5 self)
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this paper, we exploit the similarities between the Gibbs sampler and the SIS, bringing over the improvements for Gibbs sampling algorithms to the SIS setting for nonparametric Bayes problems. These improvements result in an improved sampler and help satisfy questions of Diaconis (1995) pertaining to convergence. Such an effort can see wide applications in many other problems related to dynamic systems where the SIS is useful (Berzuini et al. 1996; Liu and Chen 1996). Section 2 describes the specific model that we consider. For illustration we focus discussion on the betabinomial model, although the methods are applicable to other conjugate families. In Section 3, we describe the first generation of the SIS and Gibbs sampler in this context, and present the necessary conditional distributions upon which the techniques rely. Section 4 describes the alterations that create the second generation techniques, and provides specific algorithms for the model we consider. Section 5 presents a comparison of the techniques on a large set of data. Section 6 provides theory that ensures the proposed methods work and that is generally applicable to many other problems using importance sampling approaches. The final section presents discussion. 2 The Model
Computable bounds for geometric convergence rates of Markov chains
, 1994
"... Recent results for geometrically ergodic Markov chains show that there exist constants R ! 1; ae ! 1 such that sup jfjV j Z P n (x; dy)f(y) \Gamma Z ß(dy)f(y)j RV (x)ae n where ß is the invariant probability measure and V is any solution of the drift inequalities Z P (x; dy)V (y) V (x) ..."
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Cited by 49 (6 self)
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Recent results for geometrically ergodic Markov chains show that there exist constants R ! 1; ae ! 1 such that sup jfjV j Z P n (x; dy)f(y) \Gamma Z ß(dy)f(y)j RV (x)ae n where ß is the invariant probability measure and V is any solution of the drift inequalities Z P (x; dy)V (y) V (x) + b1l C (x) which are known to guarantee geometric convergence for ! 1; b ! 1 and a suitable small set C. In this paper we identify for the first time computable bounds on R and ae in terms of ; b and the minorizing constants which guarantee the smallness of C. In the simplest case where C is an atom ff with P (ff; ff) ffi we can choose any ae ? # where [1 \Gamma #] \Gamma1 = 1 (1 \Gamma ) 2 h 1 \Gamma + b + b 2 + i ff (b(1 \Gamma ) + b 2 ) i and i ff i 34 \Gamma 8ffi 2 ffi 3 ji b 1 \Gamma j 2 ; and we can then choose R ae=[ae \Gamma #]. The bounds for general small sets C are similar but more complex. We apply these to simple queueing models and Markov chain Mo...
Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models
 PROC. IEEE
, 2008
"... Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinitedimensional component of the hierarchical model and sample fro ..."
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Cited by 43 (5 self)
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Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinitedimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler. Since this requires imputation of an infinitedimensional process, implementation of the conditional method has relied on finite approximations. In this paper we show how to avoid such approximations by designing two novel Markov chain Monte Carlo algorithms which sample from the exact posterior distribution of quantities of interest. The approximations are avoided by the new technique of retrospective sampling. We also show how the algorithms can obtain samples from functionals of the Dirichlet process. The marginal and the conditional methods are compared and a careful simulation study is included, which involves a nonconjugate model, different datasets and prior specifications.
Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 33 (10 self)
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This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
Approximating distributions of random functionals of FergusonDirichlet priors
 Canadian Journal of Statistics
, 1998
"... The aim of this paper is to explore the possibility of approximating the FergusonDirichlet prior and the distributions of its random functionals through the simulation of random probability measures. The proposed procedure is based on the constructive definition illustrated in Sethuraman (1994) in ..."
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Cited by 28 (0 self)
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The aim of this paper is to explore the possibility of approximating the FergusonDirichlet prior and the distributions of its random functionals through the simulation of random probability measures. The proposed procedure is based on the constructive definition illustrated in Sethuraman (1994) in conjunction with the use of a random stopping rule. This allows to set in advance the closeness to the distributions of interest. The distribution of the stopping rule is derived and the practicability of the simulating procedure is discussed. Sufficient conditions for convergence of random functionals are provided. The numerical applications provided just sketch the idea of the variety of nonparametric procedures that can be easily and safely implemented in a proper Bayesian setting. Key Words: FergusonDirichlet distribution; random functionals; approximation; stopping rule. ? Dipartimento di Economia Politica e Metodi Quantitativi Universit`a di Pavia ?? Institute of Statistics and Decis...
Bayesian Inference for Semiparametric Binary Regression
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1996
"... We propose a regression model for binary response data which places no structural restrictions on the link function except monotonicity and known location and scale. Predictors enter linearly. We demonstrate Bayesian inference calculations in this model. By modifying the Dirichlet process, we obtain ..."
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Cited by 24 (2 self)
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We propose a regression model for binary response data which places no structural restrictions on the link function except monotonicity and known location and scale. Predictors enter linearly. We demonstrate Bayesian inference calculations in this model. By modifying the Dirichlet process, we obtain a natural prior measure over this semiparametric model, and we use Polya sequence theory to formulate this measure in terms of a finite number of unobserved variables. A Markov chain Monte Carlo algorithm is designed for posterior simulation, and the methodology is applied to data on radiotherapy treatments for cancer.
Computing Nonparametric Hierarchical Models
, 1998
"... Bayesian models involving Dirichlet process mixtures are at the heart of the modern nonparametric Bayesian movement. Much of the rapid development of these models in the last decade has been a direct result of advances in simulationbased computational methods. Some of the very early work in thi ..."
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Cited by 22 (2 self)
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Bayesian models involving Dirichlet process mixtures are at the heart of the modern nonparametric Bayesian movement. Much of the rapid development of these models in the last decade has been a direct result of advances in simulationbased computational methods. Some of the very early work in this area, circa 19881991, focused on the use of such nonparametric ideas and models in applications of otherwise standard hierarchical models. This chapter provides some historical review and perspective on these developments, with a prime focus on the use and integration of such nonparametric ideas in hierarchical models. We illustrate the ease with which the strict parametric assumptions common to most standard Bayesian hierarchical models can be relaxed to incorporate uncertainties about functional forms using Dirichlet process components, partly enabled by the approach to computation using MCMC methods. The resulting methology is illustrated with two examples taken from an unpub...
A proof of convergence of the Markov chain simulation method
, 1992
"... The Markov chain simulation method has been successfully used in many problems, including some that arise in Bayesian statistics. We give a selfcontained proof of the convergence of this method in general state spaces under conditions that are easy to verify. Key words and phrases: Calculation of p ..."
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Cited by 19 (2 self)
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The Markov chain simulation method has been successfully used in many problems, including some that arise in Bayesian statistics. We give a selfcontained proof of the convergence of this method in general state spaces under conditions that are easy to verify. Key words and phrases: Calculation of posterior distributions, ergodic theorem, successive substitution sampling.
Some Further Developments for StickBreaking Priors: Finite and Infinite Clustering and Classification
 Sankhya Series A
, 2003
"... this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures ..."
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Cited by 16 (0 self)
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this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures, which includes establishing a conjugacy property of their random weights to i.i.d sampling, and a characterization of the posterior for the extended stickbreaking prior under i.i.d sampling. See Section 3. These properties then lead us in Section 4 to a general characterization for the posterior of (7). In Section 5 we outline a collapsed Gibbs sampling algorithm and an i.i.d SIS (sequential importance sampling) algorithm that can be used for inference in (7). One important implication is our ability to t the posterior of (6) subject to in nite dimensional stickbreaking measures. The paper begins with a brief discussion of stickbreaking priors in Section 2